Godunov-Type Schemes for the MHD Equations

Abstract

If dissipative effects are neglected, the equations of ideal mag-netohydrodynamics (MHD) are a mathematical model for the flow of a compressible, electrically conducting fluid which is influenced by a magnetic field. They are derived from the Euler equations of fluid dynamics and the Maxwell equations and form a hyperbolic system of conservation laws. Since its behaviour is much more complicated than the Euler system’s, theoretical results and numerical schemes have not yet reached the same level as in the Euler case. This paper focuses on available approximate MHD Riemann solvers, which can be used in Godunov-type finite volume schemes: we present results of an extensive comparison, which justify the choice of the solver we use in our multidimensional code for astrophysical simulations. Moreover, we summarize some important properties of the MHD system and explain how they may influence the solutions’ structure. We conclude with two 2D applications from solar physics.